and the operator 1 is represented by an element \(e\in A_1\)  (in our case the
commutative algebra \({\mathcal {A}}\)  is \({\bf C}[L,B]\) , but the construction holds
for any commutative subalgebra
of \(\mathcal {D}\)  that contains an
element of any sufficiently large order [1]}). That the point \(P_\infty \)  which we added is smooth can be seen as follows: the affine open \(e\ne  0\)  is \(\operatorname{Spec}\big (R\big [ \frac{1}{e}\big ]_0\big )= \operatorname{Spec} {\mathcal {A}}\)  (the subscript 0 signifies the degree zero component); the affine open where \(L\ne  0\)  is \(\operatorname{Spec}\big (R\big [\frac{1}{L}\big ]_0\big )\)  and the completion of this ring in the \(e\) -adic topology is \({\bf C}[[z]]\)  if \(z\)  corresponds to \(L^iB^j/L^k\)  with \(in+jm=kn-1\)  (basically we are using \({\mathcal {L}}^{-1}\)  as a local parameter, with \({\mathcal {L}}=L^{1/n}\) ). Thus, the weight is the valuation at \(P_\infty \)  of a function in \({\mathcal {A}}\) , \(\mathcal {W}\)  is the Weierstrass semigroup and the number of gaps \(g=\frac{(m-1)(n-1)}{2}\)  is the genus of \(X\)  if there are no finite singular points.

